Local connectivity of the at certain infinitely renormalizable points∗†

Yunping Jing‡

Abstract We construct a subset consisting of infinitely renormalizable points in the Mandelbrot set. We show that Mandelbrot set is locally connected at this subset and for every point in this sub- set, corresponding infinitely renormalizable quadratic is locally connected. Since the set of Misiurewicz points is in the closure of the subset we construct, therefore, the subset is dense in the boundary of the Mandelbrot set.

Contents

1 Introduction, review, and statements of new results 237 1.1 Quadratic polynomials ...... 237 1.2 Local connectivity for quadratic Julia sets...... 240 1.3 Some basic facts about the Mandelbrot set from Douady- Hubbard ...... 244 1.4 Constructions of three-dimensional partitions...... 248 1.5 Statements of new results...... 250

2 Basic idea in our construction 252

3 The proof of our main theorem 259

∗2000 Mathematics Subject Classification. Primary 37F25 and 37F45, Secondary 37E20 †Keywords: infinitely renormalizable point, local connectivity, Yoccoz puzzle, the Mandelbrot set ‡Partially supported by NSF grants and PSC-CUNY awards and 100 men program of China

236 and Related Topics 237

1 Introduction, review, and statements of new results

1.1 Quadratic polynomials Let C and C be the and the extended complex plane (the ). Suppose f is a holomorphic map from a domain Ω ⊂ C into itself. A point z in Ω is called a periodic point of period k ≥ 1 if f ◦k(z) = z but f ◦i(z) 6= z for all 1 ≤ i < k. The number λ = (f ◦k)0(z) is called the multiplier of f at z. A periodic point of period 1 is also called a fixed point. A periodic point z of f is said to be super-attracting, attracting, neutral, or repelling if |λ| = 0, 0 < |λ| < 1, |λ| = 1, or |λ| > 1. The proof of the following theorem can be found in Milnor’s book [22, pp. 86-88].

Theorem 1.1 (Theorem of B¨ottcher). Suppose p is a super-attracting periodic point of period k of a holomorphic map f from a domain Ω into itself. There is a neighborhood W of p, a holomorphic diffeomorphism h : W → h(W ) with h(p) = 0, and a unique integer n > 1 such that

h ◦ f ◦k ◦ h−1(w) = wn for w ∈ h(W ). Furthermore, h is unique up to multiplication by an (n − 1)-st root of unity.

Consider a quadratic polynomial f(z) = az2 + bz + d. Conjugating f by an appropriate linear map h(z) = ez +s, we get h◦f ◦h−1(z) = z2 +c. So from dynamical systems point of view, quadratic polynomials form a 2 one-parameter family. We call qc(z) = z +c the Douady-Hubbard family of quadratic polynomials. When we study this family, we always deal with two kinds of sets, one is the class of sets in the phase space (the z-plane) and the other is the class of sets in the parameter space (the c-plane). In this paper, we use regular letters to denote sets in the phase space and curly letters to denote sets in the parameter space. Denote Dr = {z ∈ C | |z| < r} the disk of radius r centered 0. Let 2 −1 qc(z) = z +c be a quadratic polynomial. For r large enough, U = qc (Dr) is a simply connected domain and its closure is relatively compact in Dr. Then qc : U → V = Dr is a holomorphic, proper, degree two branched covering map. This is a model of quadratic-like maps defined by Douady and Hubbard [2] as follows: A quadratic-like map is a triple (f, U, V ) such that U and V are simply connected domains isomorphic to a disc Complex Dynamics and Related Topics 238 with U ⊂ V and such that f : U → V is a holomorphic, proper, degree two branched covering mapping. For a quadratic-like map (f, U, V ),

∞ −n Kf = ∩n=0f (U) is called the filled-in Julia set. The Julia set Jf is the boundary of Kf . Both Kf and Jf are compact. A quadratic-like map (f, U, V ) has only one critical point which we always denote as 0. Refer to [22, pp. 91-92] for a proof of the following theorem.

Theorem 1.2. The set Kf (as well as Jf ) is connected if and only if 0 is in Kf . Moreover, if 0 is not in Kf , Kf = Jf is a .

Since qc : U → V = Dr is a quadratic-like map, the filled-in Julia set of qc is the set of all points not going to infinity under forward iterates of qc. We use Kc and Jc to mean its filled-in Julia set and Julia set. For c = 0 and every r > 1, (q0,Dr,Dr2 ) is a quadratic-like map whose filled- in Julia set is K0 = D1. For any c, ∞ is a super-attracting fixed point of qc, applying Theorem 1.1, there is a unique holomorphic diffeomorphism hc (called the B¨ottcher coordinate) defined on a neighborhood B0 about 0 ∞ with hc(∞) = ∞, hc(∞) = 1 such that

−1 2 hc ◦ qc ◦ hc (z) = z , z ∈ B0.

Assume hc(B0) = C \ Dr (for a fixed large number r > 1). Let Bn = −n qc (B0). If 0 6∈ Bn, then

qc : Bn ∩ C → Bn−1 ∩ C, n ≥ 1, is unramified covering map of degree two. So we can inductively extend

hc : Bn → C \ D 1 r 2n such that −1 2 hc ◦ qc ◦ hc (z) = z z ∈ Bn. Let ◦n Bc(∞) = {z ∈ C | qc (z) → ∞ as n → ∞} be the basin of ∞. Then Kc = C \ Bc(∞). Let

+ ◦n log |qc (z)| Gc(z) = lim , n→∞ 2n Complex Dynamics and Related Topics 239

+ where log x = sup{0, log x}, be the Green function of Kc. It is a proper harmonic function whose zero set is Kc and whose critical points are bounded by Gc(0). So the B¨ottcher coordinate can be extended to

hc : Uc = {z ∈ C | Gc(z) > Gc(0)} → {z ∈ C | |z| > exp Gc(0)}. such that −1 2 hc ◦ qc ◦ hc (z) = z . −1 Moreover, Gc = log |hc|. The set Gc (log r), r > 1, is called an equipo- tential curve. If Kc is connected (equivalent to say 0 6∈ Bc(∞)), then Uc = Bc(∞) \ {∞}; if Kc is a Cantor set (equivalent to say 0 ∈ Bc(∞)), then Uc is a punctured disk bounded by an ∞ shape curve passing 0. By the definition, the Mandelbrot set M is defined as the set of parameters c such that Kc is connected. Suppose c ∈ M. Let Sr = {z ∈ C | |z| = r} be a circle of radius r > 1. Then −1 −1 sr = hc (Sr) = Gc (log r) is an equipotential curve. Note that

qc(sr) = sr2 .

−1 For every r > 1, let Ur = hc (Dr). Then Ur is a domain bounded by the equipotential curve sr. Furthermore, (qc,Ur,Ur2 ) is a quadratic-like map whose filled-in Julia set is Kc. In the rest of this paper, when we talk about a quadratic polynomial q(z) = z2 + c, it also mean a quadratic-like map (q, Ur,Ur2 ) for any r > 1. Take Eθ = {z ∈ C | |z| > 1, arg(z) = 2πθ} for 0 ≤ θ < 1. Let

−1 eθ = hc (Eθ).

Then eθ is called an of θ and

qc(eθ) = e2θ (mod 1).

An external ray is an integral curve of the gradient vector field ∇G on Bc(∞). An external ray eθ is said to land at Jc if eθ has only one limiting ◦i point at Jc. It is periodic with period m if qc (eθ) ∩ eθ = ∅ for 1 ≤ i < m ◦m and if qc (eθ) = eθ. Refer to [9, 23] for the following theorem (the reader can also find a proof in [10, pp. 199]. Complex Dynamics and Related Topics 240

2 Theorem 1.3 (Douady and Yoccoz). Suppose qc(z) = z + c is a quadratic polynomial with a connected filled-in Julia set Kc. Every repelling periodic point of qc is a landing point of finitely many periodic external rays with the same period.

Two quadratic-like maps (f, U, V ) and (g, U 0,V 0) are said to be topo- logically conjugate if there is a homeomorphism h from a neighborhood 0 Kf ⊂ X ⊂ U to a neighborhood Kg ⊂ Y ⊂ U such that

h ◦ f(z) = g ◦ h(z), z ∈ X.

If h is quasiconformal (see [1]) (resp. holomorphic), then they are quasi- conformally (resp. holomorphically) conjugate. If h can be chosen such that hz = 0 a.e. on Kf , then they are hybrid equivalent. The reader can find the proof of the following theorem in [2].

Theorem 1.4 (Douady and Hubbard). Suppose (f, U, V ) is a quadratic- like map whose Julia set Jf is connected. Then there is a unique quadratic 2 polynomial qc(z) = z + c which is hybrid equivalent to f. The following basic facts about the Julia set of a quadratic-like map (f, U, V ) will be used in the paper but the reader can check them by himself by referring to [22]. The Julia set Jf is completely invariant, −1 i.e., f(Jf ) = Jf and f (Jf ) = Jf . The Julia set Jf is perfect, i.e., 0 0 Jf = Jf , where Jf means the set of limit points of Jf . The set of all repelling periodic points Ef is dense in Jf , i.e., Ef = Jf . The limit set of −n ∞ {f (z)}n=0 is Jf for every z in V . The Julia set Jf has no interior point. If f has no attracting, super-attracting, and neutral periodic points in V , then Kf = Jf .

1.2 Local connectivity for quadratic Julia sets.

2 Consider a quadratic polynomial qc(z) = z + c whose Julia set Jc is connected. The external ray e0 is the only one fixed by qc. It lands either at a repelling or parabolic fixed point β of qc. Suppose β is repelling. From Theorem 1.3, e0 is the only external ray landing at β. So Jc \{β} is still connected. We call β the non-separating fixed point. Let α 6= β be the other fixed point of qc. If α is attracting or super-attracting, then ◦n Jc = Kc \ B(α) where B(α) = {z ∈ C | qc (z) → α as n → ∞} is the basin of α. In this case, qc is hyperbolic and Jc is a Jordan curve. If α is repelling. Then there are at least two periodic external rays landing at Complex Dynamics and Related Topics 241

α. All of them are in one cycle of period k ≥ 2. We use Λ to denote the union of this cycle of external rays. Then Λ cuts C into finitely many simply connected domains

Ω0, Ω1,..., Ωk−1.

Each domain contains points in Jc. This implies that Jc \{α} is discon- nected. We call α the separating fixed point. ◦i The point 0 is the only finite critical point of qc. Let ci = qc (0), i ≥ 1, th ∞ be the i critical value. We use CO to denote the critical orbit {ci}i=0. The critical point 0 is said to be recurrent if for any neighborhood W of 0, there is a critical value ci 6= 0 in W . We will only consider those quadratic polynomial whose critical point is recurrent and which has no neutral periodic points in this paper. These are assumed for a quadratic- like map too. Under these assumptions, qc has only repelling periodic points and Jc = Kc. Let Ur ⊂ C be the open domain bounded by an equipotential curve √ sr. For a fixed r > 1, (q, U r,Ur) is a quadratic-like map. The set Λ cuts Ur into finitely many Jordan domains. Let C0 be the closure of the one containing 0, and let B0,i be the closure of the domain containing ci for 1 ≤ i < k. We call the collection

η0 = {C0,B0,1,...,B0,k−1} the original partition about Jc. Since Λ is forward invariant under qc, the image qc(C0 ∩ Jc) (resp. q(B0,i ∩ Jc) for every 1 ≤ i < k) is the union of some sets from

η0 ∩ Jc = {C0 ∩ Jc,B0,1 ∩ Jc,...,B0,k−1 ∩ Jc}.

Each qc|B0,i is degree one, proper, and holomorphic but qc|C0 is degree two, proper, and holomorphic. −n −n For each n > 0, let αn = qc (α) and Λn = qc (Λ) (denote α0 = {α} and Λ0 = Λ). The set Λn is the union of external rays landing at points in αn. It cuts the closure of the domain U 1 into a finite number of r 2n closed Jordan domains,

ηn = {Cn,Bn,1,...,Bn,kn }.

Here we use Cn to denote the one containing 0 and Bn,1, ..., Bn,qn to denote others. Since qc(Λn) = Λn−1, qc(Cn) (resp. qc(Bn,i)) is in ηn−1. Complex Dynamics and Related Topics 242

Each qc|Bn,i is degree one, holomorphic, proper but qc|Cn is degree two, th holomorphic, and proper. We call ηn the n -partition about Jc and within it, Cn is called the critical piece. Thus we get a sequence

∞ ξ = {ηn}n=0 of nested partitions, which we call the Yoccoz partition about Jc, and a sequence of critical pieces,

0 ∈ · · · ⊆ Cn ⊆ Cn−1 ⊆ · · · ⊆ C2 ⊆ C1 ⊆ C0.

From Theorem 1.4, the sequence ξ can be also constructed similarly for any quadratic-like map whose Julia set is connected. Definition 1.1. Let (f, U, V ) be a quadratic-like map whose Julia set Jf is connected. We say it is (once) renormalizable if there is an integer n0 ≥ 2 and a domain 0 ∈ U 0 ⊂ U such that

◦n0 0 0 0 f1 = f : U → V = f1(U ) ⊂ V

0 0 0 is a quadratic-like map with connected Julia set Jf1 = J(n ,U ,V ). In 0 0 this situation, we call (f1,U ,V ) a renormalization of (f, U, V ). Other- wise, we call f non-renormalizable.

If f is renormalizable and f1 is also renormalizable, then we call f twice renormalizable. So on one can define a finitely renormalizable and infinitely renormalizable f. The relation between ξ and the renormaliz- ability of a quadratic polynomial is proved by Yoccoz (refer to [9, 23, 10]).

2 Theorem 1.5 (Yoccoz). The polynomial qc(z) = z +c is non-renormalizable ∞ if and only if ∩n=0Cn contains one point. Moreover, if qc is non-renormalizable, then the Julia set Jc is locally connected. Furthermore, we have that

Theorem 1.6 (Yoccoz). If qc is finitely renormalizable, then the Julia set Jc is locally connected. We have studied the extension of Yoccoz partitions (the first three- dimensional partition in [11]) (see §1.4) for an infinitely renormalizable quadratic polynomial firstly and used it in the study of the local connec- tivity of the Julia set of an infinitely renormalizable quadratic polynomial. More precisely, we first proved the following theorem. Complex Dynamics and Related Topics 243

Theorem 1.7 (Modulus Inequality, Jiang). Suppose f : U → V is a renormalizable quadratic-like map and 0 is not periodic. For any n0-renormalization

◦n0 0 0 0 f1 = f : U → V , n ≥ 2, we have 1 mod(U \ U 0) ≥ mod(V \ U). 2 In the theorem, mod(·) means the modulus of an annulus. Using this theorem, we showed the following corollary. The reader may refer to [11] for the concept, complex bounds.

Corollary 1.1 (Jiang). Suppose qc is an infinitely renormalizable quadratic polynomial having complex bounds. Then the Julia set Jc is locally con- nected at 0.

Furthermore, we constructed the second three-dimensional partition for an infinitely renormalizable quadratic polynomial in [11] (see §1.4), we proved the following theorem. Again, the reader may refer to [11] for the concept, unbranched.

Theorem 1.8 (Jiang). Suppose qc is a unbranched infinitely renormal- izable quadratic polynomial having complex bounds. Then the Julia set Jc is locally connected.

The unbranched condition and the complex bounds condition are im- portant in the study of local connectivity. A real infinitely renormalizable quadratic polynomial is unbranched. Many people have worked out some results about complex bounds for real infinitely renormalizable quadratic polynomials. The Feigenbaum polynomial is a real infinitely renormaliz- 2 able quadratic polynomial q∞(z) = z + t∞, t∞ real, such that

◦2i fi(z) = q∞ (z): Ui → Vi is a sequence of simple renormalizations (see [20]). Sullivan [Su] (see also [MS,Ji1]) proved that q∞ has the complex bounds. In the study of Sullivan’s result, we concluded in [15] (see also [8, Introduction] for some historic remark).

Corollary 1.2 (Jiang-Hu). The Julia set of the Feigenbaum polynomial is locally connected. Complex Dynamics and Related Topics 244

Furthermore, Levins and van Strien [17], and later, Lyubich and Yam- polsky [19] showed that any real infinitely renormalizable quadratic poly- nomial has complex bounds. Therefore,

Corollary 1.3 (Levins-van Strien and Lyubich-Yampolsky). The Julia set of a real infinitely renormalizable quadratic polynomial is locally connected.

1.3 Some basic facts about the Mandelbrot set from Douady-Hubbard Consider the Mandelbrot set M which is the compact set of all param- eters c in C such that the Julia set Jc of qc is connected. Equivalently, M consists of all parameters c such that 0 has bounded orbit (see Theo- rem 1.2). We will often identify c with the corresponding polynomial qc. A point c ∈ M (really means qc) is called hyperbolic if and only if it has a unique attracting periodic orbit. Let {z0, ··· , zp−1} be the attracting ◦p periodic orbit for a hyperbolic c and let λ(c) = qc (z0). The hyperbolic maps form an open subset in C. When c changes in one connected com- ponent H of this open set, the period p and the combinatorial type of the attracting periodic orbit are fixed. Here p is called the period of H. ◦p Moreover, λH(c) = qc (z0) maps H to D1 = {z ∈ C | |z| < 1} conformally and can be extends uniquely to a homeomorphism

λH(c): H → D1.

−1 −1 Note that H has a unique center cH = λH (0). The point rH = λH (1) is called the root of H. For example, the component H0 bounded by the cardioid is the hyperbolic component√ of c such that qc has an attracting fixed point. Then λH0 (c) = 1 − 1 − 4c. The center is 0, the root is 1/4, and λ−1(e2πiθ) = e2πiθ(2 − e2πiθ)/4. H0 Consider a hyperbolic component H of period p ≥ 1 and the corre- m 2πi 0 0 sponding λH(c). For each r ∈ ∂H such that λH(r) = e p ,(m, p ) = 1, it is the root of a hyperbolic component H0 of period pp0. (Note that in the case m = 0 and p0 = 1, H0 = H and in all other cases, H0 6= H.) Here H0 is called the satellite of H with an internal angle m/p0 and denoted as H0 = H ∗ H(m/p0). For each c ∈ C, let

hc : Uc = {z ∈ C | Gc(z) > Gc(0)} → {z ∈ C | |z| > exp (Gc(0))} Complex Dynamics and Related Topics 245

be the B¨ottcher coordinate from §1.1, where Gc is the Green function for Kc. Then −1 2 hc ◦ qc ◦ hc (z) = z . and Gc = log |hc|. For c ∈ C \M, c ∈ Uc and thus we have hc(c). Theorem 1.9 (Douady-Hubbard). The map

ΦM(c) = hc(c): C \ M → C \ D1 is a . Thus M is connected. The equipotential curve of radius r > 1 of M is

−1 Sr = ΦM ({c ∈ C | |c| = r}) and the external ray of angle 0 ≤ θ < 1 of M is

−1 Eθ = ΦM ({c ∈ C | |c| > 1 and arg(c) = 2πθ}).

For example, E0 = (1/4, ∞) and E1/2 = (−∞, −2). An external ray Eθ is said to land at M if it has only one limiting point at M. Both of E0 and E1/2 land at M (E0 lands at 0 and E1/2 lands at −2). Furthermore,

1) every external ray Eθ of rational angle θ = m/p lands at a point in M;

0 2) if θ = m/p with (m, p) = 1 and p = 2p , then Eθ lands at a point c ∈ M such that the critical orbit CO of qc is preperiodic (such a c is called a ). Conversely, every Misiurewicz point is a landing point of an external ray Eθ of angle θ = m/p with (m, p) = 1 and p = 2p0; 3) For θ = m/p with (m, p) = 1 and p = 2p0, let c be the landing (Misiurewicz) point of Eθ, the external ray eθ in the z-plane for qc lands at its critical value of c (see [28] for some similarity between M around a Misiurewicz c and the corresponding Jc around c); 4) let H be a hyperbolic component of period p > 1, there are exactly − two external rays Eθ− and Eθ+ land at the root rH where θ = m−/(2p − 1) and θ+ = m+/(2p − 1) for 1 ≤ m− < m+ < 2p − 1.

Consider the component H0 bounded by the cardioid and a satellite H(m/p) = H∗H0(m/p) of angle m/p,(m, p) = 1. Let Eθ− and Eθ+ be two + external rays landing at the root rH(m/p). Then the closure of Eθ− ∪ Eθ cuts C into two connected components. Let Wm/p be the one containing H(m/p). See [7] for the following theorem. Complex Dynamics and Related Topics 246

Theorem 1.10 (Goldberg-Milnor). For any c ∈ Wm/p, consider qc. There are exactly p external rays in the z-plane that land at the separating i + fixed point αc of qc. These p external rays have 2 θ , i = 0, ··· , p− 1. Moreover, c is in the domain bounded by two external rays of angle θ− and θ+. The main conjecture in this direction is that the Mandelbrot set M is locally connected. From the Carath´eodory theorem, if M is locally −1 connected, then ΦM can be extended to a comtinuous map from C \ D1 to C \M, thus every external ray lands at a unique point in M. By applying the Douady-Hubbard map ΦM, one can transfer Yoccoz ∞ ∞ partition ξ = {ηn}n=1 to a partition Ξ = {Θn}n=0 around all finitely renormalizable points in the parameter space. Using this para- partition, Yoccoz further proved the following theorem (refer to [9, 24]). Theorem 1.11 (Yoocoz). The Mandelbrot set M is locally connected at every finitely renormalizable point.

For indifferent points c (i.e., qc has a neutral periodic point), Yoccoz proved that the Mandelbrot set is locally connected at these points. The key step in his proof is the Yoccoz inequality as follows (refer to [9]). (For rational indiferent points, one needs to argue more, see [25] or [29] for a clarification.) Consider a monic polynomial P (z) of degree d whose Julia set is connected. Let p be a repelling fixed point of P and let λ = P 0(p). Suppose there are totally m0 external rays of P landing at p. Label these external rays in cyclical order. Suppose the ith external ray is mapped to the [(i + k0) (mod m0)]th external ray. Let r = gcd(m0, k0). Then there are r cycles of external rays landing at p. Let m0 = rm and k0 = rk, (m, k) = 1. The Yoccoz inequality says that there is a branch τ of log λ satisfying <τ rm ≥ . k 2 2 log d |τ − 2πi m | In other words, τ belongs to the closed disc of radius (log d)/(rm) tangent to the imaginary axis at 2πik/m. The Yoccoz inequality reveals a rela- tion between the analytic derivative τ and the combinatorial derivative 2πik/m of P at a fixed point. Concluding from Yoccoz’ results, the Mandelbrot set M is locally connected at those points c such that qc are not infinitely renormalizable. Therefore, the study of the local connectivity of the Mandelbrot set M is concentrated at all infinitely renormalizable points. Complex Dynamics and Related Topics 247

By applying the Yoccoz inequality, Douady-Hubbard constructed a generic subset of infinitely renormalizable quadratic polynomials whose Julia set is non-locally connected but M is locally connected at every point in this subset. They used a method called tuning (or called unrenor- malization) in this construction. The tuning can be described roughly as follows. Consider a hyperbolic component H of period p. There are two exter- nal rays Eθ−(H) and Eθ+(H) land at its root rH. The set Eθ−(H) ∪ Eθ+(H) ∪ {rH} cuts C into two domains. Denote the one containing H as W(H). Then W(H) contains a small copy M(H) of the Mandelbrot set M such that H is the image of H0. For every c ∈ M(H), qc is renormalizable and, more precisely, there is a simple renormalization

◦m(H) qc : U[c] → V [c].

Let H0 be a satellite of H of period p0. We can similarly construct W(H0) and M(H0) for H0 but we will denote them as W(H ∗ H0) and M(H ∗ H0). (Without having confusion, we also denote them as W(p ∗ p0) and M(p ∗ p0)). For every c ∈ W(p ∗ p0),

◦m(p) qc : U[c] → V [c] is renormalizable and has a simple renormalization

◦m(p)p0 0 0 qc : U [c] ⊂ U[c] → V [c] ⊂ V [c].

This processing is called tuning. Starting from the component H0 bounded by the cardioid and a sequence of positive integers, p0, p1, ···, Douady and Hubbard constructed a sequence of nested tuning sets ˜ Wp0∗p1∗···∗pn = W(p0 ∗ p1 ∗ · · · ∗ pn) ∩ M.

Using the Yoccoz inequality, they showed that if pn tends to ∞ fast ˜ enough (i.e., pn >> pn−1), the diameter d(Wp0∗p1∗···∗pn ) tends to zero as n goes to infinity, therefore, M is locally connected at the intersection point ∞ ˜ c ∈ ∩n=0Wp0∗p1∗···∗pn .

Furthermore, qc is infinitely renormalizable whose Julia set is not locally connected (see [23]) due to the fast growth of pn. In this paper, we will study those infinitely renormalizable points whose Julia sets are locally connected. Complex Dynamics and Related Topics 248

1.4 Constructions of three-dimensional partitions.

2 Take an infinitely renormalizable quadratic polynomial qc(z) = z + c. Let U and V be domains bounded by equipotential curves such that 0 f0 = qc : U → V is a quadratic-like map. Let α0 = α, β0 = β, ηn = ηn, 0 0 0 0 ∞ 0 ξ = ξ, Cn = Cn,Λn = Λn, and Λ∞ = ∪n=0Λn be the same as §1.2. Let n1 ≥ 0 and k1 ≥ 2 be two integers such that

◦k1 0 0 ∞ 0 f1 = f0 : Ck1+n1 → Cn1 ⊂ N(J1, 1) where J1 = ∩i=0Ci .

Suppose β1 and α1 are the non-separating and separating fixed points of f1, i.e., J1 \{β1} is still connected and J1 \{α1} is not. The points β1 and α1 are also repelling periodic points of qc. There are at least two, 1 but a finite number, external rays of qc landing at α1. Let Λ0 be the 1 1 0 union of external rays landing at α1. Then Λ0 cuts V0 = Cn1 into a finite 1 number of closed domains. Let η0 be the collection of these domains. Let 1 −n 1 1 1 −n 1 Λn = f1 (Λ0) for any n > 0. Then Λn cuts Vn = f1 (V0 ) into a finite 1 number of closed domains. Let ηn be the collection of these domains. 1 1 ∞ The sequence ξ = {ηn}n=0 is a sequence of nested partitions about J1. We call it the first partition. (We also call ξ0 the 0th partition.) Let 1 ∞ 1 Λ∞ = ∪n=0Λn. 1 1 The domain Cn ∈ ηn containing 0 is called the critical piece. It is 1 clear the restriction f1 to Cn is degree two branched covering map but to 1 all other domains in ηn are degree one. Let

∞ 1 J2 = ∩n=0Cn.

There are two integers n2 ≥ 0 and k2 ≥ 2 such that

◦k2 1 1 f2 = f1 : Cn2+k2 → Cn2

1 is a degree two branched cover map and such that Cn2 ⊂ N(J2, 1/2). Inductively, for every i ≥ 2, suppose we have already constructed

f = f ◦ki : Ci−1 → Ci−1. i i−1 ni+ki ni

Let βi and αi be the non-separating and separating fixed points of fi; i.e., Ji \{βi} is still connected and Ji \{αi} is not. The points βi and αi are also repelling periodic points of qc. There are at least two, but a finite i number, external rays of qc landing at αi. Let Λ0 be the union of external i i i−1 rays landing at αi. Then Λ0 cuts V0 = Cni into finitely many closed Complex Dynamics and Related Topics 249

i i −n i domains. Let η0 be the collection of these domains. Let Λn = fi (Λ0) i i −n i for any n > 0. Then Λn cuts Vn = fi (V0 ) into finitely many closed i i domains. Let ηn be the collection of these domains. The domain Cn i i in ηn containing 0 is called the critical piece in ηn. It is clear that fi i i restricted to all domains but Cn are bijective and fi|Cn is a degree two branched covering map. Let

∞ i Ji+1 = ∩n=0Cn.

There are two integers ni+1 ≥ 0, ki+1 ≥ 2 such that

◦ki+1 i i fi+1 = fi : Cni+1+ki+1 → Cni+1

i is a degree two branched cover map and such that Cni+1 ⊂ N(Ji+1, 1/(i+ i i ∞ 1)). Let ξ = {ηn}n=0. It is the sequence of nested partitions about Ji. We call it the ith partition.

◦ki+1 0 0 Remark 1.1. For any ki+1-renormalization fi+1 = fi : U → V of i 0 fi : Ui → Vi, we have an integer n > 0 such that Cn ⊂ V ∩N(Ji+1, 1/(i+ 1)) and f = f ◦ki+1 : Ci → Ci is a degree two branched covering i+1 i n+ki+1 n map. We will still use ξi to mean ξi ∩ Ci . Therefore, (U ,V ) n+ki+1 i+1 i+1 ◦ki+1 can be an arbitrary domains such that fi+1 = fi : Ui+1 → Vi+1 is a ki+1-renormalization of fi : Ui → Vi. Qi Let mi = j=1 ki, 1 ≤ i < ∞. We have thus constructed a most natural infinite sequence of simple renormalizations,

◦mi ∞ {fi = qc : Ui → Vi}i=1,

i ∞ ∞ and the nested-nested sequence {ξ }i=0 of partitions about {Ji}i=0 (where i ∞ J0 = Jc). Then Ξ = {ξ }i=0 is our first three-dimensional partition about Jc. Now we construct our second three-dimensional partition Υ about Jc. Denote by κ1 the first partition which will be constructed as follows: 0 0 ∞ 0 0 0 Consider ξ = {ηn}n=0 in Ξ. Take Ck(0) ∈ ηk(0) ∈ ξ where k(0) = n1 +k1. 0 0 Put all domains in ηk(0)+1 which are the preimages of Ck(0) under qc into 0c 0 0c κ1 and let ηk(0)+1 be the rest of the domains. Consider ηk(0)+2 ∩ ηk(0)+1 0 consisting of all domains in ηk(0)+2 which are subdomains of the domains 0c 0 0c in ηk(0)+1. Put all domains in ηk(0)+2 ∩ ηk(0)+1 which are the preimages of 0 ◦2 0c Ck(0) under qc into κ1 and let ηk(0)+2 be the rest of the domains. Suppose we already have η0c for s ≥ 2. Consider η0 ∩ η0c consisting of k(0)+s k(0)+s+1 k0+s Complex Dynamics and Related Topics 250

0 0c all domains in ηk(0)+s+1 which are subdomains of the domains in ηk(0)+s. 0 0c 0 Put all domains in ηk(0)+s+1 ∩ ηk(0)+s which are the preimages of Ck(0) ◦(s+1) 0c under qc into κ1 and let ηk(0)+s+1 be the rest of the domains. Thus we can construct the partition κ1 inductively. This partition covers points 0 in Jc minus all points not entering the interior of Ck(0) under all forward iterations of qc. 1 1 ∞ 1 1 1 Next consider the ξ = {ηn}n=0 in Ξ. Take Ck(1) ∈ ηk(1) ∈ ξ where k(1) = n2 + k2. We can use similar arguments to those in the previous paragraph by considering f : C0 → C0 (to replacing q : U → V ) 1 k(0) k(0)−k1 c 0 to get a partition κ1,1 in Ck(0). Then we use all iterations of qc to pull back this partition following κ1 to get a partition κ2. It is a sub-partition of κ1 and covers points in Jc minus all points not entering the interior of 1 Ck(1) under iterations of qc. th Suppose we have already constructed the (j − 1) partition κj−1 for j j ∞ j j j j ≥ 2. Consider the partition ξ = {ηn}n=0 in Ξ. Take Ck(j) ∈ ηk(j) ∈ ξ j−1 where k(j) = nj+1 + kj+1. Similarly, by considering fj : Ck(j−1) → Cj−1 , we get a partition κ in Cj−1 . Then we use all backward k(j−1)−kj j,1 k(j−1) iterations of fj−1 to pull back this partition following κj−1 to get a par- j−2 tition κj,2 in Ck(j−2) and all backward iterations of fj−2 to pull back this j−3 partition following κj−1 to get a partition κj,3 in Ck(j−3), and so on to obtain a partition κj = κj,j in V . It is a sub-partition of κj−1 and covers j points in the Julia set minus all points not entering the interior of Ck(j) under forward iterations of qc. By the induction, we have a sequence of nested partitions ∞ Υ = {κj}j=1 ∞ which covers points in Jc \ Γ. Then Υ = {κj}j=1 is our second three- dimensional partition about Jc.

1.5 Statements of new results. Yoccoz partitions can be transferred completely to the parameter space naturally by using the Douady-Hubbard map ΦM. But it is more diffi- cult to transfer three-dimensional partitions to the parameter space com- pletely. The difficulty is around those Feigenbaum-like points. In this paper, we will partially transfer three-dimensional partitions to the pa- rameter space. We will construct a subset of the Mandelbrot set such that (1) this subset consists of infinitely renormalizable points, (2) this subset is dense on the boundary of the Mandelbrot set, (3) we can trans- Complex Dynamics and Related Topics 251 fer three-dimensional partitions for infinitely renormalizable quadratic polynomials in this subset to a partition around this subset set in the parameter space, and (4) the partition is good enough to study the local connectivity of this subset in the Mandelbrot set. More precisely, we prove in this paper that

Theorem 1.12 (Main Theorem). Suppose c is a Misiurewicz point. Then there is a subset A(c) ⊂ M such that

(i) c is a limit point of A(c),

0 2 0 (ii) for every c ∈ A(c), qc0 (z) = z + c is an unbranched infinitely renormalizable quadratic polynomial having complex bounds and the Julia set Jc0 is locally connected, and (iii) the Mandelbrot set M is locally connected at every point c0 ∈ A(c).

Since the set B of Misiurewicz points is dense on the boundary of M, we have

Corollary 1.4. The set A = ∪c∈BA(c) is dense on the boundary ∂M. In the proof of Theorem 1.12, we will use the following classical the- orem in (refer to [16]).

Theorem 1.13 (Rouch´e’sTheorem). Suppose D is a domain in the complex plane C. Let γ ⊂ D be a closed path homologous to 0 and assume that γ has an interior. Let f and g be analytic on D, and

|f(z) − g(z)| < |f(z)|, z ∈ γ.

Then f and g have the same number of zeros in the interior of γ.

The rest of the paper is arranged as follows. To have a better expla- nation to our idea, we first study a special Misiurewicz point −2 in the Mandelbrot set and prove in §2 the following theorem.

Theorem 1.14 (Special Case). There is a subset A(−2) ⊂ M con- sisting of infinitely renormalizable points such that −2 is a limit point of A(−2) and the Mandelbrot set M is locally connected at A(−2), More- over, for every c ∈ A(−2), the Julia set Jc of qc is locally connected. Then by combining the proof of Theorem 1.14, we give a complete proof of Theorem 1.12. Complex Dynamics and Related Topics 252

2 Basic idea in our construction

2 Let q−2(z) = z − 2. Its Julia set J2 is [−2, 2]. It has the non-separating fixed point β(−2) = 2 and the separating fixed point α(−2), i.e., J−2 \ {β(−2)} is still connected and J−2 \{α(2)} is disconnected. To use notations clearly, we use the letter G to denote a set in the phase space and the Letter Λ to denote the same set but in the B¨ottcher coordinate. Let G0,l(−2) be the closure of the union of two external rays landing at α(−2). Let h−2 be the B¨otteche coordinate for q−2, i.e.,

−1 2 h−2 ◦ q−2 ◦ h−2(z) = z .

Then h−2 maps G0,l(−2) to two straight rays in C\D1 (which have angles 1/3 and 2/3). We use Λ0,l to denote the closure of the union of these two straight rays. Let

LH = {z = x + yi ∈ C | x < 0} and RH = {z = x + yi ∈ C | x > 0}

2 be the left and right half planes. Consider q0(z) = z . The restriction q0|(C \ (−∞, 0]) has two inverse branches

gl,0 : C \ (−∞, 0] → LH and gr,0 : C \ (−∞, 0] → RH.

n+1 n+1 Let Λn,r = gr,0 (Λ0,l) and Λn+1,l = gl,0 (Λn,r) for n = 0, 1, ···. It is easy to see that Λn,r tends to the ray of angle 0 and Λn,l tends to the ray of angle 1/2. Let

∞ Λ = [∪n=0(Λn,r ∪ Λn,l)] ∪ ((−∞, −1] ∪ [1, ∞)).

−1 Let G(−2) = h−2(Λ). Then it is the union of external rays for q−2 landing at the set

∞ n n A(−2) = [∪n=0(gr,−2(α(−2)) ∪ gl,−2(α(−2)))] ∪ {−2, 2}.

−1 −1 We also denote Gn,r(−2) = h−2(Λn,r) and Gn,l(−2) = h−2(Λn,l). For c ∈ W1/2, let G0,l(c) be the closure of union of two external rays landing at its separating fixed point α(c). Let α(c) be another preimage of α(c). Let G0,r(c) be the closure of union of two external rays landing at α(c). Then G0,l(c) ∪ C0,r(c) cuts C into three domains. One, which we call Er(c), contains the non-separating fixed point β(c) of qc. The other, Complex Dynamics and Related Topics 253

D0 G0,l (−2) s(−2)

B1 D1

B2 D 2 β β α 0 α

Figure 1: Basic idea in our construction in the dynamical plane.

which we call El(c), contains the other preimage β(c) of β(c) under qc. The restriction qc|(El,c(c) ∪ Er,c)(c) has two inverse branches

gl,c : E → El(c) and gr,c : E → Er(c).

2 Let hc be the B¨ottcher coordinate for qc(z) = z + c, i.e.,

−1 2 hc ◦ qc ◦ hc = z .

−1 Let G(c) = hc (Λ). Then it is the union of external rays for qc landing at the set

∞ n n A(c) = [∪n=0(gr,c(α(c)) ∪ gl,c(α(c)))] ∪ {β(c), β(c)}.

−1 −1 We also denote Gn,r(c) = hc (Λn,r) and Gn,l(c) = hc (Λn,l). Let −1 φc = hc ◦ h−2.

When restricted to G(−2), φc conjugates q−2|G(−2) to qc|G(c). Let U(−2) be a fixed domain bounded by an equipotential curve s(−2) for q−2. Then q−2 : U(−2) → V (−2) is a quadratic-like map where V (−2) = q−2(U(−2)). The set G0,l(−2)∪G0,r(−2) cuts U(−2) into three disjoint domains. Denote D0(−2) the closure of the one containing 0 (see Figure 1). ◦n ◦n Let Dn(−2) = gr,−2(D0(−2)) and let Bn(−2) = gl,−2(Dn−1(−2)) for n ≥ 1 (see Figure 1). Since 2 is an expanding fixed point of q−2 and q(−2) = 2, the diameter diam(Bn) of Bn tends to zero exponentially as n goes to infinity. Let 0 6∈ U0 be a small neighborhood about −2 in the parameter space such that diam(U0) ≤ 1 and such that the corresponding Complex Dynamics and Related Topics 254

graph G(c) for qc exists for c in U0. For c in U0, let φc : G(−2) → G(c) be the conjugacy from q−2 to qc. Let sc = φc(s) be the corresponding equipo- tential curve for qc. Let U(c) be the closure of the domain bounded by sc. Similarly, G0,l(c) ∪ G0,r(c) cuts U(c) into three disjoint domains. Denote ◦n D0(c) be the closure of the one containing 0. Let Dn(c) = gr,c(D0(c)) and let Bn(c) = gl,c(Dn−1(c)) for n ≥ 1. Note that β(c) and α(c) are the non-separating and separating fixed points of qc and β(c) is the other inverse image of β(c) under qc. Then Bn(c) and Dn(c) tend to β(c) and β(c), respectively, as n goes to infinity. Since β(c) is an expanding fixed 0 point of qc and since there is a constant µ > 1 such that |qc(β(c))| ≥ µ for all c in U0, the diameter diam(Bn(c)) tends to 0 uniformly on U0 and the set Bn(c) approaches to β(c) uniformly on U0 as n goes to infinity. Let Wn = {c | c ∈ Bn(c)}.

Then Wn, for n = 1, 2, ···, give a partition in the parameter space. Let f(c) = qc(0) − β(c) and g(c) = qc(0) − x for x ∈ Bn(c). Suppose γ = ∂U0 is a closed path homologous to 0 in U0 such that

m = min = min |qc(0) − β(c)| > 0. c∈∂U0 c∈∂U0

Because Bn(c) approaches β(c) uniformly on U0 as n goes to infinity, there is an integer N0 > 0 such that for n ≥ N0,

|f(c) − g(c)| = |β(c) − x| < m ≤ |f(c)|, c ∈ ∂U0.

Since the equation f(c) = qc(0) − β(c) = 0 has a unique solution −2 in U0, Theorem 1.13 (Rouch´e’sTheorem) implies that g(c) = qc(0) − x = 0 has a unique solution, which is in U0, for any x in Bn(c) for n ≥ N0. Therefore Wn ⊆ U0 for n ≥ N0. So diam(Wn) ≤ 1 for n ≥ N0 (see Figure 2). ˜ Lemma 2.1. The intersection Mn = Wn ∩ M for n ≥ N0 is connected.

Proof. The boundary Bn(c) consists of four curves Gn,l(c), Gn−1,l(c), and −n some pieces of the equipotential curve sn(c) = qc (s(c)). Note that n+1 n qc (Gn,l(c)) = G0,l(c) and qc (Gn−1,r(c)) = G0,l(c). The domain Wn is bounded by the closure of two external rays

Gn = {c | c ∈ Gn,l(c)}, Gn−1 = {c | c ∈ Gn−1(c)} and some pieces of the equipotential curve

Sn = {c | c ∈ sn(c)}. Complex Dynamics and Related Topics 255

U0

Mn

M Mn+2 n+1

Wn+2

Wn+1

Wn

Figure 2: Basic idea in our construction in the parameter space.

Each of intersections Gn ∩ M and Gn−1 ∩ M consists only one point, denoted as an and an−1. The closure of Gn (or Gn−1) in the extended complex plane is a topological curve isomorphic to a circle and it inter- sects M only at an (or an−1). Let us still denote these extended curves as Gn and Gn−1. Then Gn and Gn−1 cut C into three domains. The one on an side is called Xn and the one on an−1 side is called Xn−1. ˜ If Mn is disconnected. Let U and V be two non-empty domains such that U ∩ V = ∅ and such that ˜ ˜ ˜ (U ∩ Mn) ∪ (V ∩ Mn) = Mn.

Assume an ∈ U and an−1 ∈ V (other cases can be proved similarly). Let ˜ ˜ ˜ ˜ U = U ∪ Xn and V = V ∪ Xn−1. Then U and V are two non-empty domains such that U˜ ∩ V˜ = ∅ and such that

(U˜ ∩ M) ∪ (V˜ ∩ M) = M.

This would say that M is disconnected. This contradiction implies that ˜ Mn must be connected.

Proof of Theorem 1.14 (Special Case). For each Wn where n ≥ N0, since −1 c ∈ Bn(c), Cn(c) = qc (Bn(c)) is the closure of a connected and simply connected domain which contains 0 and which is a sub-domain of D0(c). Let ◦(n+1) ˚ ˚ fn,c = qc : Cn(c) → D0(c). Complex Dynamics and Related Topics 256

D (c) 0 F n,c

Cn (c)

Figure 3: Construction of a quadratic-like map by renormalization.

Then it is a quadratic-like map (see Figure 3). Furthermore, since diam(Bn(c)) tends to zero as n goes to infinity uniformly on U0, for N0 large enough we ˚ can have that the modulus mod(An(c)) ≥ 1 where An(c) = D0(c)\Cn(c). Moreover, the family ˚ ˚ {fn,c : Cn(c) → D0(c); c ∈ Wn} is a full family, so Wn contains a copy Mn of the Mandelbrot set M

(refer to [23, pp. 102-106], ). For c ∈ Mn, the Julia set Jfn,c of fn,c : ˚ ˚ ∞ Cn(c) → D0(c) is connected. Therefore, for c ∈ A1 = ∪n≥N0 Mn, qc is once renormalizable.

For a fixed integer i0 ≥ N0, consider Wi0 and Mi0 ; there is a param- eter c ∈ M such that f = f : C˚ = C˚ (c ) → D˚ = D˚ (c ) is i0 i0 i0 i0,ci0 i0 i0 i0 i0 0 i0 2 ˚ hybrid equivalent to q−2(z) = z − 2. The quadratic-like map fi0 : Ci0 → ˚ Di0 has the non-separate fixed point and the separate fixed point, which we simple denote as β and α. Let β be another pre-image of β under fi0 . Let G be the closure of the union of two external rays for q landing ci0 at α. Then f (G) = G and G ∩ J = {α}. Let G˜ = f −1(G). Then G˜ i0 fi0 i0 cuts Ci0 into three domains. Denote Di00 be the one containing 0. Let

β ∈ Ei00 and β ∈ Ei01 be the components of the closure of Ci0 \ Di00. Let gi00 and gi01 be the inverses of fi0 |Ei00 and fi0 |Ei01. Let

◦n Di0n = gi01(Di00) and Bi0n = gi00(Di0(n−1)) for n ≥ 1. Again by the B¨ottcher coordinates h and h , we can similarly ci0 c ˜ prove that β, β, α, and G, G, and ∂Ci0 are preserved for c close to ci0 . Similar to the argument in the above, we can find a small neighborhood Complex Dynamics and Related Topics 257

Ui0 about ci0 with diam(Ui0 ) ≤ 1/2 such that the corresponding domains

Bi0 (c) and Di0 (c) can be constructed for qc for c ∈ Ui0 . Let

Wi0n = {c ∈ C | fi0,c(0) ∈ Bi0n(c)}.

Then {Wi0n}i0≥1,n≥1 give a sub-partition of {Wn}n≥1.

The diameter diam(Bi0n(c)) tends to zero uniformly on Ui0 and the set

Bi0n(c) approaches to β(c) uniformly on Ui0 as n goes to infinity. Suppose

Ui0 is simply connected and the boundary curve γ = ∂Ui0 is a closed path homologous to 0 in Ui0 . Since the equation fi0,c(0) − βi0 (c) = 0 has a unique solution ci0 (following Thurston’s theorem for critically finite rational maps (see [5]) and also refer to [4]), m = min |f (0) − c∈∂Ui0 i0,c

βi0 (c)| > 0. Let f(c) = fi0,c(0) − βi0 (c) and g(c) = fi0,c(0) − x for x ∈ Bi0n(c). There is an integer Ni0 > 0 such that for n ≥ Ni0 ,

|f(c) − g(c)| = |βi0 (c) − x| < m ≤ |f(c)|, c ∈ ∂Ui0 .

Theorem 1.13 (Rouch´e’sTheorem) now implies that g(c) = fi0,c(0) − x =

0 has a unique solution in Ui0 for any x in Bi0n(c) for n ≥ Ni0 . Therefore

Wi0n ⊆ Ui0 for n ≥ Ni0 . So diam(Wi0n) ≤ 1/2 for n ≥ Ni0 . ˜ Similar to the proof of Lemma 2.1, we have Mi0n = Wi0 ∩ M is connected for n ≥ Ni0 . For each c in Wi0n, n ≥ Ni0 , let Ci0n(c) = −1 fi0,c(Bi0n(c)). Then ◦(n+1) ˚ ˚ fi0n,c = fi0,c : Ci0n(c) → Di00(c)

is a quadratic-like map. Furthermore, since diam(Bi0n(c)) tends to zero as n goes to infinity uniformly on Ui0 , for Ni0 large enough we can have the ˚ modulus mod(Ai0n(c)) ≥ 1 where Ai0n(c) = Di00(c) \ Ci0n(c). Moreover, ˚ ˚ {fi0n,c : Ci0n(c) → Di00(c) | c ∈ Wi0n}

is a full family. Thus Wi0n contains a copy Mi0n of the Mandelbrot set M (refer to [23, pp. 102-106]). For c ∈ A = ∪ ∪ M , the Julia 2 i0≥N0 i1≥Ni0 i0i1 ˚ ˚ set of fi0n,c : Ci0n(c) → Di00(c) is connected, so qc is twice renormalizable. We use the induction to complete the construction of our subset A(−2) around −2. Suppose we have constructed Ww where w = i0i1 . . . ik−1 and i0 ≥ N0, i1 ≥ Ni1 , ..., ik−1 ≥ Ni0i1...ik−2 . Let v = i0 . . . ik−2. There ˚ ˚ ˚ is a parameter cw ∈ Mw such that fw = fw,cw : Cw = Cw(cw) → Dw = ˚ 2 Dv0(cw) is hybrid equivalent to q−2(z) = z − 2. The quadratic-like map ˚ ˚ fw : Cw → Dw has the non-separate fixed point βw and the separate fixed Complex Dynamics and Related Topics 258

point αw. Let Gw be the closure of the union of two external rays for ˜ −1 ˜ qcw which land at αw. Let Gw = fw (Gw). Then Gw cuts Cw into three domains. Let Dw0 be the one containing 0. Denote βw be another preim- age of βw under fw. Let βw ∈ Ew0 and βw ∈ Ew1 be the components of the closure of Cw \ Dw0. Let gw0 and gw1 be the inverses of fw|Ew0 and fw|Ew1. Let

◦n Dwn = gw1(Dw0) and Bwn = gw0(Dw(n−1))

for n ≥ 1. By the B¨ottcher coordinates hcw and hc, βw, βw, αw, and Gw, ˜ Gw, and ∂Cw are preserved for c close to cw. We can find a small neigh- k borhood Uw about cw with diam(Uw) ≤ 1/2 such that the corresponding domains Dwn(c) and Bwn(c) can be constructed for qc, c ∈ Uw. Let

Wwn = {c ∈ C | fw,c(0) ∈ Bwn(c)}, n = 1, 2, ··· .

They give a sub-partition of {Ww}. The diameter diam(Bwn(c)) tends to zero uniformly on Uw and the set Bwn(c) approaches to βw(c) uniformly on Uw as n goes to infinity. Suppose Uw is simply connected and the boundary curve ∂Uw is a closed path homologous to 0. Since the equation fw,c(0) − βw(c) = 0 has a unique solution cw (following from Thurston’s theorem for critically finite rational maps (see [5]) and also refer to [4]),

m = min |fw,c(0) − βw(c)| > 0. c∈γ

Let f(c) = fw,c(0) − βw(c) and g(c) = fw,c(0) − x for x ∈ Bwn(c). There is an integer Nw > 0 such that for n ≥ Nw,

|f(c) − g(c)| = |βw(c) − x| < m ≤ |f(c)|, c ∈ ∂Uw.

Now Theorem 1.13 (Rouch´e’sTheorem) implies that g(c) = fw,c(0)−x = 0 has a unique solution in Uw for any x in Bwn(c) and n ≥ Nw. Therefore k Wwn ⊆ Uw for n ≥ Nw. So diam(Wwn) ≤ 1/2 for n ≥ Nw. ˜ Similar to the proof of Lemma 2.1, we have Mwn = Wwn ∩ M for n ≥ Nw is connected. For each c in Wwn where n ≥ Nw, let Cwn(c) = −1 fw,c(Bwn(c)). Then

◦(n+1) ˚ ˚ fwn,c = fw,c : Cwn(c) → Dw0(c) is a quadratic-like map. Furthermore, since diam(Bwn(c)) tends to zero as n goes to infinity uniformly on Uw, for Nw large enough we can have Complex Dynamics and Related Topics 259

˚ that the modulus mod(Awn(c)) ≥ 1 where Awn(c) = Dw0(c) \ Cwn(c). Moreover, ˚ ˚ {fwn,c : Cwn(c) → Dw0(c) | c ∈ Wwn} ˜ is a full family. Therefore, Wwn contains a copy Mwn ⊆ Mwn of M (refer to [23, pp. 102-106]). For c ∈ Ak+1 = ∪w ∪ik≥Nw Mwik , qc is (k +1)-times renormalizable where w = i0i1 . . . ik−1 runs over all sequences of integers of length k. We thus construct a three-dimensional partition

∞ {Wi0...ik }k=0

∞ about the subset A(−2) = ∩k=1Ak. For each c ∈ A(−2), qc is infinitely renormalizable. Furthermore, −2 is a limit point of A(−2). For each c ∈ A(−2), there is a correspond- ∞ ing sequence w∞ = i0i1 . . . ik ... of integers such that {c} = ∩k=0Wi0...ik . ˜ ∞ Since Mi0...ik = Wi0...ik ∩ M is connected, {Wi0...ik }k=0 is a basis of con- nected neighborhoods of M at c. In other words, M is locally con- nected at c. Moreover, from our construction, qc is unbranched and has complex bounds. Therefore the Julia set Jc is locally connected from Theorem 1.8.

3 The proof of our main theorem

Proof of Theorem 1.12 (Main Theorem). Let c0 ∈ M be a Misiurewicz point and Jc0 be its Julia set. Then there is the smallest integer m ≥ 1 ◦m such that p = qc0 (0) is a repelling periodic point of qc0 of period k ≥ 1. We start the construction of our subset A(c0) and a three-dimensional partition {Ww | w = i0i1 ··· in, n = 0, 1, · · ·} around it as follows.

Let α be the separating fixed point of qc0 . Let G be the closure of ∞ the union of external rays landing at α. Let ξ = {ηn}n=0 be the Yoccoz partition about Jc0 (see §1.2). Let

p ∈ · · · ⊆ Dn(p) ⊆ Dn−1(p) ⊆ · · · ⊆ D1(p) ⊆ D0(p) be a p-end, that means that p ∈ Dn(p) ⊆ Dn−1(p) and Dn(p) ∈ ηn. Let

c0 ∈ · · · ⊆ En(c0) ⊆ En−1(c0) ⊆ · · · ⊆ E1(c0) ⊆ E0(c0) be a c0-end, this means that c0 ∈ En(p) ⊆ En−1(p) and En(p) ∈ ηn. ◦(m−1) We have qc0 (En+m−1(c0)) = Dn(p). Since the diameter diam(Dn(p)) tends to zero as n → ∞ and since p is a repelling periodic point, we can Complex Dynamics and Related Topics 260

◦k 0 find an integer l ≥ m such that |(qc0 ) (x)| ≥ λ > 1 for all x ∈ Dl(p) and ◦(m−1) such that qc0 : El+m−1(c0) → Dl(p) is a homeomorphism. Let t ≥ 0 ◦t be the integer such that f = qc0 : Dl(p) → Cr0 is a homeomorphism, where Cr0 is the domain containing 0 in ηr0 , r0 ≥ 0. There is an integer −1 r > r0 such that r + t > l and such that B0 = f (Cr ∩ Dl(p)) does not ◦t contain p. Thus qc0 : B0 → Cr is a homeomorphism. Define

³ ´−1 ◦nk Bn = qc0 |Dl+nk(p) (B0) ⊆ Dl+nk(p)

◦(q+nk) for n ≥ 1. Note that Bn is in ηr+q+nk. Then qc0 : Bn → Cr is a homeomorphism. By the B¨ottcher coordinates hc0 and hc, α, G, p, Cr, Dn, and Bn, for n ≥ 0, are all preserved for c close to c0. Therefore they can be constructed for qc as long as c close to c0 as we did for −2. Let U0 be a neighborhood about c0 with diam(U0) ≤ 1 such that the corresponding α(c), G0(c), p(c), Cr(c), Dn(c), and Bn(c), for n ≥ 0, are all preserved for c ∈ U0. As n goes to infinity, the diameter diam(Bn(c)) tends to zero uniformly on U0 and the set Bn(c) approaches to p(c) uniformly on U0. Let m Wn = Wn(c0) = {c ∈ C | qc (0) ∈ Bn(c)}, n ≥ 1. They give a partition in the parameter space. Suppose U0 is simply connected and the boundary γ is a closed path m homologous to 0 in U0. Since the equation qc (0) − p(c) = 0 has a unique solution c0 in U0 (following Thurston’s Theorem for critically finite ratio- ◦m nal maps (see [5]) and also refer to [4]), m = minc∈γ |qc (0) − p(c)| > 0. ◦m ◦m Let f(c) = qc (0) − p(c) and g(c) = qc (0) − x for x ∈ Bn(c). Since Bn(c) approaches p(c) uniformly on U0 as n goes to infinity, there is an integer N0 = N0(c0) > 0 such that for n ≥ N0,

|f(c) − g(c)| = |p(c) − x| < m ≤ |f(c)|, x ∈ ∂U0.

◦m Theorem 1.13 (Rouch´e’sTheorem) now implies that g(c) = qc (0)−x = 0 has a unique solution, which is in U0, for any x in Bn(c) and n ≥ N0. Therefore Wn ⊆ U0 for n ≥ N0. So diam(Wn) ≤ 1 for n ≥ N0.A similar argument to the proof of Lemma 2.1 implies that, for n ≥ N0, ˜ Mn = M ∩ Wn is connected (see Figure 4). For any c ∈ Wn, n ≥ N0, let Rn(c) be the pre-image of Bn(c) under the map ◦(m−1) qc0 : El+m−1(c) → Dl(p, c) Complex Dynamics and Related Topics 261

Figure 4: A small copy of the Mandelbrot set and hairs around it.

−1 and let Cm+r+q+nk(c) = qc (Rn(c)). Then Cm+r+q+nk(c) is the closure of a domain which contains 0 and which is in ηm+r+q+nk. Hence ◦(q+nk+m) ˚ ˚ fn,c = qc : Cm+r+q+nk(c) → Cr(c) is a quadratic-like map. Furthermore, since diam(Bn(c)) tends to zero as n goes to infinity uniformly on U0, for N0 large enough we can have the ˚ modulus mod(An(c)) ≥ 1 where An(c) = Cr(c)\Cm+r+q+nk(c). Moreover, ˚ ˚ {fn,c : Cm+r+q+nk(c) → Cr(c) | c ∈ Wn} is a full family. Thus Wn contains a copy Mn = Mn(c0) of the Man- ˜ delbrot set M (refer to [23, pp. 102-106]). Note that Mn ⊆ Mn and ˜ Mn \Mn is usually not empty (see Figure 4). For any c ∈ A1(c0) =

∪n≥N0 Mn, qc is once renormalizable. Now following almost the same arguments in the proof of Theo- rem 1.14, We can construct a subset A(c0) and a three-dimensional par- tition {Ww(c0) | w = i0 ··· in, n = 0, 1, · · ·} about it such that c0 is a limit point of A(c0) and such that every c ∈ A(c0) is infinitely renormalizable at which M is locally connected. Furthermore, qc is unbranched and has complex bounds from our con- struction. Therefore the Julia set Jc is locally connected following The- orem 1.8. It completes the proof. Remark 3.1. Eckmann and Epstein [6] and Douady-Hubbard [4] have ˜ estimated the size of Mn. Since Mn \Mn contains hairs (see Figure 4) Complex Dynamics and Related Topics 262 which may destroy the local connectivity of M, we must estimate the size ˜ of Mn. This is the key point in the proof. Remark 3.2. Lyubich [18] constructed another subset consisting of in- finitely renormalizable points such that M is locally connected at every point in this subset. A point in his subset must satisfies several compli- cate conditions. His idea is more close to Douady and Hubbard’s idea. A point in our susbet may not satisfy those conditions. Our idea is different and simple and originated in [11, 12, 13, 14].

Remark 3.3. In our proof of Theorem 1.12, the use of Rouch´e’sTheorem is interesting. Actually, A three-dimensional partition in the parameter space can be constructed around more general infinitely renormalizable points (those points are not eventually (2, 2, 2, ···)-renormalizable) just like we did in the proof. However finding a tool to replace Rouch´e’sThe- orem in the proof is an interesting problem. A candidate is Slodkowski’s Theorem [26] for holomorphic motions. We would like to explore this in the further research.

References

[1] L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Mathematical studies, Vol. 10, D. Van Nostrand Co. Inc., Toronto- New York-London, 1966.

[2] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like maps, Ann. Sci. Ecole.´ Norm. Sup (4) Vol. 18, no. 2, 1985, 287–344.

[3] A. Douady and J. H. Hubbard, It´erationdes polynˆomesquadratiques complexes, C. R. Acad. Sci. Paris, Vol. 294, 1982, 123-126.

[4] A. Douady and J. H. Hubbard, Etude´ dynamique des polynˆomes complexes I& II, Publ. Math. Orsay, 1984-85.

[5] A. Douady and J. H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math., Vol. 171, 1993, 263-297.

[6] J. Eckmann and H. Epstein, Scaling of Mandelbrot sets by critical point preperiodicity, Comm. Math. Phys., Vol. 101 (1985). Complex Dynamics and Related Topics 263

[7] L. Goldberg and J. Milnor, Fixed points of polynomial maps, Part II: fixed point portaits, Ann. Sci. Ec.¨ Norm. Sup., Vol. 26, 1993, 51-98.

[8] J. Hu and Y. Jiang, The Julia set of Feigenbaum quadratic poly- nomial, Proceedings of the International Conference in Honor of Professor Shantao Liao, World Scientific Publishing Co. Inc., River Edge, NJ, 1999, 99-124.

[9] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J. -C. Yoccoz. Topological Methods in Modern Mathematics, Publish or Perish, Houston, TX, 1993, 467–511.

[10] Y. Jiang, Renormalization and Geometry in One-Dimensional and Complex Dynamics, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.

[11] Y. Jiang, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc., Vol. 352, no 11 (2000), 5077-5091.

[12] Y. Jiang, Markov partitions and Feigenbaum-like mappings, Comm. Math. Phys., Vol. 171 (1995), 351-363.

[13] Y. Jiang, Infinitely renormalizable quadratic Julia sets, preprint, 1993.

[14] Y. Jiang, On the Mandelbrot set at infinitely renormalizable points, MSRI preprint 95-63, Berkeley.

[15] Y. Jiang and J. Hu, The Julia set of Feigenbaum quadratic polyno- mial is locally connected, preprint, 1993.

[16] S. Lang, Complex Analysis, 4th Edition, Springer-Verlag, New York, 1999.

[17] G. Levin and S. van Strien, Local connectivity of the Julia sets of real polynomials, Ann. of Math (2). Vol. 147, no. 3, 1998, 471–541.

[18] M. Lyubich, Geometry of quadratic polynomials: Moduli, rigidity, and local connectivity, Stony Brook IMS Preprint 1993/9.

[19] M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: complex bounds for real maps, Ann. Inst. Fourier (Grenoble), Vol. 47, no. 4, 1997. 1219–1255. Complex Dynamics and Related Topics 264

[20] C. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, Vol. 135, Princeton Univ. Press, Princeton, NJ, 1994.

[21] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1993.

[22] J. Milnor, Dynamics in One Complex Variable, 2nd Edition, Vieweg & Sohn, 2000.

[23] J. Milnor, Local connectivity of Julia sets: expository lectures, in The Mandelbrot Set, Theme and Variations, Cambridge University Press, Cambrigde, 2000, 67-116.

[24] P. Rœsch, Holomorphic motions and puzzles (following M. Shishikura), in The Mandelbrot Set, Theme and Variations, Cam- bridge University Press, Cambrigde, 2000, 117-131.

[25] D. Schleicher, Rational parameter rays of the Mandelbrot set, Stony Brook I.M.S Preprint 1997/#13.

[26] Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc., Vol. 111 (1991), 347-355.

[27] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society Centennial Publica- tions, Vol. 2, Mathematics into the Twenty-First Century, AMS, Providence, RI, 1988, 417–466.

[28] Tan Lei, Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys., Vol. 134 (1990), 587–617.

[29] Tan Lei, Local properties of the Mandelbrot set at parabolic points, in The Mandelbrot set, Theme and Variations, Cambridge Univer- sity Press, 2000, 133-160.

Yunping Jiang, Department of Mathematics, Queens College of CUNY, Flushing, NY 11365, USA and Department of Mathematics, CUNY Grad- uate Center, 365 Fifth Ave., New York, NY 10016, USA & the 100 Man Program in the Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China Email: [email protected] Homepage: www.qc.edu/˜yunqc